Some functional-difference equations solvable in finitary functions

Gorin, E.

doi:10.1090/S1061-0022-07-00973-9

Some functional-difference equations solvable in finitary functions
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by E. A. Gorin
Translated by: S. V. Kislyakov

St. Petersburg Math. J. 18 (2007), 779-796

DOI: https://doi.org/10.1090/S1061-0022-07-00973-9

Published electronically: August 9, 2007

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Abstract:

The following equation is considered: $q(-i\partial /\partial x)u(x)=(f*u)(Ax)$, where $q$ is a polynomial with complex coefficients, $f$ is a compactly supported distribution, and $A:\mathbb {R}^n\to \mathbb {R}^n$ is a linear operator whose complexification has no spectrum in the closed unit disk. It turns out that this equation has a (smooth) solution $u(x)$ with compact support. In the one-dimensional case, this problem was treated earlier in detail by V. A. Rvachev and V. L. Rvachev and their numerous students.

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